Tetrahedralization of non-conformal three-dimensional mixed element meshes

ABSTRACT

A dynamic directory and tetrahedralization method. The dynamic directory of degree of freedom data for elements in a non-conformal mixed-element mesh includes elements subdividable into tetrahedral, in which a respective degree of freedom value is stored for each element, wherein the degree of freedom value is current as element subdivision proceeds. The tetrahedralization method includes providing a non-conformal mixed element mesh comprising elements subdividable into tetrahedra, identifying respective degree of freedom values for the elements in the mesh, and performing element subdivision based on the degree of freedom values of elements in the mesh.

The present invention is a continuation of U.S. patent application Ser.No. 09/942,419 filed Aug. 30, 2001, the disclosure of which is expresslyincorporated by reference herein in its entirety.

FIELD OF THE INVENTION

The invention generally relates to tetrahedralization of meshes, moreparticularly, to three dimensional tetrahedralization of non-conformalmixed-element meshes.

DISCUSSION OF BACKGROUND INFORMATION

Real-world problems (such as movement of impurities in siliconsemiconductor components) sometimes can be modeled and simulated oncomputers. Many simulators provide output as mixed-element meshes, i.e.,meshes including tetrahedral and non-tetrahedral elements. Tetrahedraare three-dimensional shapes with four vertices and four triangularfaces. Examples of non-tetrahedral shapes are blocks, pyramids andprisms.

It has been desired to further process the simulator output, such as torun it through visualization or testing software programs. However, somesuch software programs require input to be tetrahedral meshes, andcannot accommodate non-tetrahedral meshes. Thus, techniques were soughtand developed for tetrahdedralizing mixed-element meshes. A particularchallenge has been to tetrahedralize a “non-conformal” mixed-elementmesh, i.e., one in which triangular element faces can abut quadrilateralelement faces in the mesh. Three dimensional tetrahedralization is awell known problem. See, e.g., G. Albertelli, R. Crawfis, “Efficientsubdivision of finite element datasets into consistent tetrahedra,” IEEE(1997); J. Dompierre, P. Labbe, M. Vallet, R. Camarero, “How tosubdivide pyramids, prisms and hexahedra into tetrahedra,” Centre derecherche en calcul applique (CERCA), presented at the 8.sup.thInternational Meshing Roundtable, South Lake Tahoe, Calif. (October1999), pages 195-204.

To subdivide non-tetrahedra (e.g., prisms and hexahedra) into tetrahedrareduces to selecting how to split each quadrilateral face with adiagonal. For example, prisms have three quadrilateral faces; eachquadrilateral face can be split by a diagonal in two different ways;therefore there are eight possible face-splitting combinations for eachprism. Only six of these can be subdivided into three tetrahedra withoutSteiner points; the other two require Steiner points, and produce eighttetrahedra. Similarly, hexahedra have six quadrilateral faces, resultingin 2⁶=64 possible face-splitting combinations. Only 46 of these can besubdivided into five or six tetrahedra without Steiner points; the other18 are indivisible.

When a non-tetrahedral shape is indivisible, to subdivide it requirescreation of a “Steiner point”. Additional Steiner points increase thenumber of nodes and elements in the mesh. Steiner points in the inputmeshes interfere with the speed and the convergence behavior of softwareprocessing the meshes. Conventional approaches to subdividingnon-conformal mixed-element meshes into tetrahedral meshes have neededto add relatively many undesirable “Steiner points” to generate thetetrahedral mesh. A Steiner point is a node added to the elementinterior, usually at its center.

In an initially conformal mesh, there are no initial constraints on howthese quadrilateral faces must be split, except that abuttingquadrilateral faces must be split in a consistent manner (i.e., thediagonal selected for the face from one element must match the diagonalselected for the face of the adjacent element which shares that face).The Dompierre algorithm can be used in such a non-tetrahedral conformalmesh to subdivide all elements into tetrahedra. J. Dompierre et al,supra. While the Dompierre et al. strategy is very elegant and fast, andrequires no Steiner points, it cannot be applied to subdivide initiallynon-conformal meshes. As Dompierre observes, the problem of subdividingmeshes occurs, e.g., in computer graphics where certain meshes must besubdivided into tetrahedra to use efficient algorithms for volumerendering, iso-contouring and particle advection.

In “non-conformal” meshes, abutting triangular faces initially constrainthe choice of diagonal for some quadrilateral faces, so that Dompierrecannot be used. These initial constraints can directly or indirectlyresult in “overconstrained” prisms or hexahedra, i.e., those that cannotbe subdivided into tetrahedra without adding Steiner points. In aninitial mesh having prisms, hexahedra and tetrahedra, to subdivide themesh, determinations are needed for splitting each quadrilateral facewith a diagonal. Not all possible diagonal choices, for each prism andhexahedra, have a corresponding tetrahedralization. Non-conformalitiesin an initial mesh constrain the subdivision of some prisms andhexahedra. In some cases, an element may initially be, or may becomeduring subdivision, overconstrained, making a consistent subdivisionimpossible. In such a case, a Steiner point must be used to subdivide anoverconstrained element. Steiner points are undesirable, because theyincrease the mesh size and reduce the local mesh quality. In FIG. 1, anexample of a non-conformal mesh situation is shown, with two tetrahedraabutting a hexahedron's quadrilateral face.

In FIGS. 2A-L, quadrilateral face constraint patterns are shown. FIGS.2A-D are four divisible cases without a three-diagonal vertex. FIGS.2E-L are alternating bit pattern cases, with no matched diagonals, noprisms possible; two of the cases are divisible (FIGS. 2K-L), six casesare not (FIGS. 2E-3J). The two divisible cases (FIGS. 2K-L) give fivetetrahedra.

For subdividing elements in a non-conformal mixed element mesh,generally, certain random and non-random subdivisions have beenproposed. Random subdivision subdivides a randomly selected firstelement, and each subsequent selection of an element to be subdividedalso has no particular geometric basis. For example, a list of elementsmay be randomly generated, and subdivided in that random order.

Certain non-random approaches have been proposed, that take into accountsome geometrical feature, e.g., the strategy of Albertelli, et al. forsubdividing an initially non-conformal mixed element mesh, i.e., tobegin with subdividing the elements in random order, and use a“depth-first-search” subdivision approach. Albertelli's methodology maybe imagined as starting randomly with any element, and then moving to anadjacent neighboring element for the next subdivision, each time movingto a neighboring element. While the Albertelli algorithm can subdividethese non-conformal meshes, the resulting subdivided mesh contains anundesirably large number of Steiner points.

Steiner points in tetrahedral mesh input cause interference and slowsimulation time (such as by increasing node and element count; andintroducing abrupt 2:1 mesh scale changes which degrade convergence).Minimizing added nodes (Steiner points) in tetrahedralizing meshes isimportant for making those meshes more useable by software applicationsrequiring tetrahedral mesh input.

SUMMARY OF THE INVENTION

The present invention tetrahedralizes a “non-conformal” mixed-elementmesh, i.e., one in which triangular element faces can abut quadrilateralelement faces in the mesh, with a minimal number of Steiner points. Theinvention can be applied to non-conformal meshes, and vastly reduces theneed for additional nodes, without the need for local transformations onthe existing mesh. The invention further does not suffer from drawbacksof Albertelli.

In order to accomplish these and other objects of the invention, thepresent invention in a preferred embodiment provides atetrahedralization method, comprising at least the steps of: providing anon-conformal mixed element mesh comprising elements subdividable intotetrahedra, and identifying respective degree of freedom values for theelements in the mesh; and performing element subdivision based on thedegree of freedom values of elements in the mesh. In a particularlypreferred embodiment of the invention tetrahedralized output isobtained.

The invention additionally provides, as another preferred embodiment, adynamic directory of degree of freedom data for elements in anon-conformal mixed-element mesh comprising elements subdividable intotetrahedra. The inventive dynamic directory comprises a respectivedegree of freedom value for each element, wherein the degree of freedomvalue is current as element subdivision proceeds.

In a particularly preferred embodiment, strategic element subdivision isapplied, such as look-ahead, or, when multiple subdivisions of anelement are possible, applying a subdivision pattern closest tosatisfying Dompierre “global numbering” criteria.

In the invention, degree of freedom for elements in the mesh is treatedas nonstatic. The invention advantageously provides for post-subdivisionupdating of the degree of freedom data, such as updating after eachelement subdivision or updating after a batch of elements have beensubdivided.

Additionally, the invention provides for breadth-first-searchsubdivision, such as generating nearest newly-constrained elements andsubdividing all nearest newly-constrained elements before subdividing aneighbor of a nearest newly-constrained element.

Another preferred embodiment of the invention provides atetrahedralizing filter, comprising: a receiver for data defined on anon-conformal mixed element mesh comprising elements subdividable intotetrahedra, a processor for the mesh data, wherein the processordynamically associates individual to-be-subdivided elements in the meshwith a degree of freedom value in an element-by-element degree offreedom directory; and an element subdivider that discriminates onwhether to initiate subdivision or hold subdivision based on the degreeof freedom directory, with subdivision priority to relativelymost-constrained to-be-subdivided elements. The inventive filteroptionally includes a subdivision strategizer and/or an updatabledirectory.

The filter may include a breadth-first-search subdivider that generatesnearest newly-constrained elements and subdividing all nearestnewly-constrained elements before subdividing a neighbor of a nearestnewly-constrained element. A preferred example of breadth-first-searchsubdivision that may be used in the present invention is as follows:initially select an arbitrary element to subdivide, then subdivide in abreadth-first manner in which after subdividing the first element, eachneighbor of the first element is visited and subdivided before visitingsecond nearest neighbors of the first element; after visiting all theneighbors of the first element, one of the neighbors is arbitrarilyselected and then all of its neighbors are visited, after which anotherneighbor of the first element is selected and then all of its neighborsare visited.

In another preferred embodiment, the invention provides tetrahedralizedoutput data produced by providing a non-conformal mixed element meshcomprising elements subdividable into tetrahedra, and generating datadefining respective degree of freedom values for the elements in themesh; and performing element subdivision based on the degree of freedomvalues of elements in the mesh, wherein the degree of freedom data isdynamically updated. Advantageously, the invention providestetrahedralized output including a minimal number of, or no, Steinerpoints.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, aspects and advantages will be betterunderstood from the following detailed description of the preferredembodiments of the invention with reference to the drawings, in which:

FIG. 1 is a diagram of an exemplary non-conformal mesh situation, withtwo tetrahedra abutting a hexahedra's quadrilateral face.

FIGS. 2A-L include diagrams of a subset of the 64 possible quadrilateralface constraint patterns for a block-shaped element.

FIG. 3 is a diagram showing global numbering of a three-dimensionalmesh.

FIG. 4 is a flow-chart of subdivision of elements in a mesh according tothe present invention.

FIG. 5 is an exemplary recursive marking algorithm according to theinvention.

FIG. 6 is a flow-chart of an application using tetrahedralizationaccording to the present invention.

FIG. 7 illustrates a tetrahedralization filter according to theinvention.

DETAILED DESCRIPTION OF THE EMBODIMENTS OF THE INVENTION

The present invention in tetrahedralizing an initial non-conformal meshperforms element subdivision based on each element's degree of freedom(“DOF”), i.e., the number of possible remaining diagonal choices thatproduce a tetrahedralizable element (such as stored DOF data). Mostpreferably, DOF is treated non-statically, such as updating DOF data forremaining to-be-subdivided elements after an element subdivision.Additionally, and also most preferably, the subdivision of elementshaving DOF greater than one is strategic, i.e., not random.

An initial non-conformal mesh that may be tetrahedralized according tothe present invention is any mesh having a combination of elements suchas prisms, hexahedra and/or tetrahedra. In the inventive methods,products and processes, the mesh may be three-dimensional. As examplesof sources of initial non-conformal meshes may be mentioned output fromsimulation tools such as Taurus sold by Avanti. The mesh preferably isglobally numbered with a number (preferably an integer) assigned to eachnode. Global numbering is known to those skilled in the art. An exampleof global numbering of a three-dimensional mesh may be seen with regardto FIG. 3, in which a simple mesh is shown, consisting of blocks 100 and101 and pyramid 102.

A to-be-subdivided element in the mesh is an element that has a non-zeroDOF. An element that has zero DOF, i.e., is over-constrained, cannot besubdivided without the addition of a Steiner point. The initial meshmay, but is not required to, contain one or more elements that areindivisible (i.e., have zero DOF).

DOF determination of a to-be-subdivided element preferably is treated asnonstatic, such as by performing dynamic DOF updating. An example ofdynamic DOF updating is, before any subdivision occurs, an initialelement-by-element DOF determination followed by, after a subdivision ofan element has occurred, a DOF re-determination for some or allremaining to-be-divided elements, such as dividing an element followedby updating the DOF on all its neighbors before dividing anotherelement.

A DOF determination of a to-be-subdivided element results in an integervalue, and the DOF of each element is a function of the particular mesh.For an initial non-conformal mixed-element mesh, it is believed that DOFpossibilities are 0, 1-10, 13, 14, 23 and 46. In a most preferredexample of DOF determination, DOF of each element in the mesh iscomputed before any subdivision begins, and the DOF values are stored asDOF data, and treated as non-static values subject to later updating.That is, it will be appreciated that the DOF value of each element isnot necessarily static, and may change as a result of a subdivisionoperation. When an element is subdivided, the other elements in the meshwhose DOF could change are the neighbors of the subdivided element.Thus, it is preferred to redetermine DOF for each neighbor element of asubdivided element (which when performed as a computer-assistedcalculation is not a particularly “expensive” operation), withoutdetermining all remaining elements in the mesh.

Element subdivision based on each element DOF preferably includessubdivision proceeding batchwise from most to least constrained (i.e.,lowest to highest DOF) elements. An exemplary subdivision schemeaccording to the invention may be appreciated with reference to FIG. 4,in which an initial step is provided of setting (110) a threshold DOF(such as 1 or another relatively-low DOF such as 4) and applyingrecursive subdivision 111 (such as a recursive subdivision algorithmcomprising identifying and subdividing undivided elements with DOF lessthan or equal to the threshold DOF). The threshold DOF may be anynon-zero integer, however, most preferably it is selected from the groupconsisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 23 and 46. Theinitial setting of the threshold DOF may be any non-zero integer, butpreferably it is a relatively-low DOF (such as 1, 2, 3, 4) and mostpreferably it is 1. In addition to identifying the relationship of anelement's DOF with respect to the threshold DOF, disposition of theelement is provided, e.g., a command that if in processing a list ofelements in the mesh, an element having a DOF equal to or less than thethreshold DOF is encountered, subdivision of that element may proceed,while if an element having a DOF greater than the threshold DOF isencountered, return to the list of elements occurs. Preferably, incarrying out identification of elements equal to or less than thethreshold DOF, stored data for DOF values for the elements in the meshare used.

Once an element is determined to be eligible to proceed to subdivision,element subdivision may proceed for those elements. For example,subdivision may proceed simultaneously of elements having DOF equal toor less than the threshold DOF, or subdivision may proceedelement-by-element of elements having such a relatively-low DOF. By wayof example, if the threshold DOF is 4, elements having DOF of 4, 3, 2and 1 would be eligible for subdivision in the batch where the DOFthreshold is 4. Of the list of elements having DOF equal to or less than4, those elements may be taken up for subdivision in random order. Inprocessing the list of elements, thus, where the threshold DOF, a firstelement subdivided could have DOF of 3, the next element could have DOFof 2, a next element could have DOF of 1, a next element could have DOFof 4, etc., but an element having a DOF of 5 or more would not be takenup for subdividing in that batch. An exemplary example of applyingrecursive subdivision 111 in FIG. 4 may comprise calling MARKQ (i.e.,mark quad) of FIG. 5, i.e., running the recursive subdivision algorithmof FIG. 5 starting with a particular element.

Recursive subdivision (111) may be of a batch of elements, or of asingle element. Element subdivision begins with a batch of relativelymost-constrained elements or a relatively-most constrained element. Apreferred example of recursive subdivision (111) is calling mark qaccording to the recursive subdivision algorithm in FIG. 5, i.e.,running the subdivision algorithm on a particular element in the inputmesh. A threshold test for DOF may be built into MARKQ as a MARKQsubroutine.

In subdividing (111), preferably strategic subdivision features and/orbox-in avoidance features are included, e.g., not subdividing an elementhaving a DOF greater than the threshold DOF (i.e., holding relativelyunconstrained elements for later subdivision); look-ahead (such asconfirming that a current subdivision choice results in neighborelements which can be subdivided without Steiner points); drawingdiagonals according to Dompierre rules (such as consistently selecting adiagonal through the lowest-numbered node of the to-be-dividedquadrilateral face of the to-be-divided element), use of breadth-firstsubdivision strategy (e.g., divide all neighbors of current elementsbefore moving on to second nearest neighbors, etc.), etc.

At the conclusion of a subdivision for a first batch of DOF threshold,the mesh usually contains one or more subdivided elements. It will beappreciated that during the subdivision (111) process, the DOF of theremaining to-be-divided elements in the mesh is dynamic, and can beexpected to have changed from before the subdivision. Thus, theinvention provides for optional post-subdivision DOF recalculation orre-determination. Applying post-subdivision DOF recalculation is highlyadvantageous and particularly preferred. DOF recalculation may beperformed at any time after at least one element has been subdivided.Preferably, after element subdivision (111), DOF recalculation forremaining to-be-subdivided elements previously determined to be abovethe DOF threshold is performed, to determine if any element should beadded to the current subdivision batch. As examples may be mentioned DOFrecalculation after subdivision of a single element, or DOFrecalculation after completion of a batch of elements.

After subdivision (111), including any optional DOF recalculation,inquiry (112) is made whether any non-zero elements remain unsubdivided,and if so, threshold DOF is reset (110) and undivided elements withnon-zero DOF less than or equal to the new threshold DOF are identifiedand the subdivision (111) process is repeated.

The cycle is repeated until inquiry (112) of whether any non-zero DOFelements remain is negative. At that point, zero DOF (i.e.,over-constrained) elements, if any, are processed (113), e.g., byintroduction of Steiner points. It will be appreciated that the presentinvention provides a real-time solution for tetrahedralizing a mesh toprovide a minimal number of Steiner points, without necessarily entirelyavoiding Steiner points. The present invention includes a heuristicsolution that balances the need for quickly obtaining a tetrahedralizedmesh with relatively few Steiner points with the vast computation timeand resources that would be required to try each possible subdivisionscheme, starting from different elements, applying differentsubdivisions, etc. The present invention thus quickly and reliablyprovides a tetrahedralized mesh that is satisfactorily input intosoftware applications requiring a tetrahedralized mesh.

It will be appreciated that in effecting the element subdivision (111),when the DOF is one, no choice is available and only one subdivision ispossible, however, when the DOF is greater than one, subdivision choicesare available. Preferably, in selecting between subdivision choices, astrategic, i.e., a non-random, approach is applied. A strategicsubdivision approach permits, but does not require, attempting allsubdivision choices.

An exemplary strategic subdivision approach to subdivision (111)includes lookahead confirmation whenever more than one DOF is availablefor subdividing an element. Look-ahead may be to one or more levels,with looking ahead one level being preferred. Generally, looking-aheadmore than one level is thought to be relatively too cumbersome, byslowing processing time without achieving benefits worth the burden. Thelook-ahead comprises confirming which current subdivision choice resultsin neighbor elements which can be subdivided without Steiner points.

Another strategic approach to subdivision (111) preferably includes,when multiple ways to subdivide the quadrilaterals in an element arepossible, selecting a subdivision pattern that comes closest tosatisfying the Dompierre “global numbering” criteria.

The present invention thus provides an enhanced recursive markingalgorithm for an initially non-conformal mesh, with an exemplaryalgorithm being shown in FIG. 5, in which Dompierre pattern scoring 120,one-level look-ahead 126 and a breadth-first marking sequence 127 areused. Initially a threshold DOF is set, and then the algorithm is called(114) for that DOF on every undivided element in the mesh. A screeninginquiry (114 a) is whether an element EO (which, particularly as theprocess begins, may be randomly selected from a list of elements in themesh) has been subdivided already; if so, return (114 b) is provided tothe list of elements in the mesh; if not, inquiry is made whether theDOF of the element EO is greater than the threshold DOF (115). To answerinquiry (115), a table of DOF values for each element in the mesh may beconsulted and a comparison made between the DOF value for the element EOand the threshold DOF. Preferably the table of DOF values is a dynamicdirectory of DOF values updated as of the last subdivision of an elementin the mesh.

If the result of the threshold DOF/element DOF comparison inquiry (115)is that the element EO DOF is greater than the threshold DOF, thenreturn (116) to the list of to-be-divided elements is provided. If theDOF of the element is less than or equal to the threshold DOF, then theelement is passed on for treatment by the subdivision algorithm (117),in which inquiry is made for each valid pattern (i.e., a subdivisionpattern which produces a subdividable element, i.e., a collection ofdiagonal choices) P0, from best to worst a la Dompierre. Inquiry (117)is an example of global numbering feature of the present invention.

In FIG. 5, selecting a subdivision pattern for the element proceeds byordering the patterns with individual scores according to the number ofdiagonals which pass through the lowest numbered node in theirquadrilateral faces (i.e., Dompierre's strategy for conformal meshsubdivision). The patterns can be sequenced, such that the choose ofvalid pattern is not random. Work on one element at a time, screening(118) of each valid subdivision pattern is performed from theperspective of the element's neighbors. Valid subdivision patterns areprocessed in order of most desirable to least desirable (117). Inscreening each valid subdivision pattern for an element according to theperspective of neighboring elements, the reference point is of oneneighboring element at a time. The references to done/not done in FIG. 5refer to whether the loop has been completed for each neighboringelement. Once a loop has been made through all valid patterns from theviewpoint of a neighboring element, the loop is done and one steplook-ahead may now be performed for the tentative subdivision pattern.One-step look-ahead shown in FIG. 5 for a tentative subdivision patternfor an element looks at every neighbor of the element being tentativelysubdivided and tentatively updates the DOF pattern for all thoseneighbors. With that tentatively updated neighbor DOF information,inquiry (119) is made whether the DOF of the neighbor element would bezero. If the result of the inquiry (119) is that the neighboring elementwould not be over-constrained, the inquiry passes to the next neighborelement. If a neighbor element that is over-constrained is encountered,return is provided to inquiry (117) and the next-best valid patternaccording to Dompierre is processed. In FIG. 5, one-step look aheadaccording to the invention comprises the neighbor-by-neighbor (118)inquiry of whether the neighbor is over-constrained (119).

When a valid pattern is being tested in the look-ahead from theperspective of the neighbor elements, if all neighbor elements areprocessed and none are found to be over-constrained as a result of thepattern, the done path is followed and the pattern is accepted (122) forthe element, i.e., a successful pattern has been found. The algorithm ofFIG. 5 proceeds in a breadth-first manner for each neighbor EN, ofcalling and applying the algorithm. Any element that already has beenactually subdivided is not subdivided again, tentatively or actually. Atentative subdivision may be confirmed as an actual subdivision, or maybe bypassed in favor of a better tentative subdivision which thenbecomes the actual subdivision.

As shown in FIG. 5, for each neighbor (123), MARKQ is called (124). Theprocess is not done for an element until all neighbors of the elementhave been processed. When done with all neighbors of an element, return(125) is provided. In FIG. 5, breadth-first-search according to theinvention comprises the neighbor-by-neighbor (123) calling of MARKQ(124), i.e., the recursive algorithm.

Returning to block (117) on FIG. 5, the case is addressed where allDompierre patterns have been processed and none have been foundacceptable. In such a case, the done branch is followed and the patternwith the least number of over-constrained neighbors is used (121) andthe process continues to block (123).

Practicing the methods of the present invention advantageously resultsin reducing the number of Steiner points needed to tetrahedralize atypical non-conformal mixed element mesh compared to practicing theAlbertelli technique. The reductions achieved by the present inventionmay be of a factor of two or more, such as a factor of about 16 or 20.By reducing the number of Steiner points, the present inventionadvantageously provides both the speed and the convergence behavior ofsoftware which uses the resulting tetrahedral mesh.

An exemplary process according to the invention may be seen with respectto FIG. 6, which depicts a process in which a data generator 130 (suchas finite element or control volume software) produces data defined onan initial non-conformal mixed element mesh 131. The data 131 areentered into a tetrahedralization filter 132 which outputs data 133defined on a tetrahedral mesh. The data 133 are entered into software134 receiving data defined on a tetrahedral mesh. A preferred example ofthe filter is an input filter for a visualization program such as theData Explorer visualization program, or for three-dimensionalprocess/device simulation, e.g., simulation of a silicon device havingan impurity as data defined on a non-conformal mixed element mesh,tetrahedralizing the data, and processing the tetrahedralized data,e.g., as to the movement of the impurity under various conditions ofcharge, temperature, etc.

An exemplary embodiment of a tetrahedralization filter 139 isillustrated in FIG. 7, in which a receiver 140 receives the data definedon the non-conformal mixed element mesh, which is composed of elementssubdividable into tetrahedral. A processor 141 for the mesh data isprovided to dynamically associate individual to-be-subdivided elementsin the mesh with a degree of freedom value in an element-by-elementdegree of freedom directory 142. An element subdivider 143 discriminateson whether to initiate subdivision or hold subdivision based on thedegree of freedom directory, in which subdivision priority is torelatively most-constrained to-be-subdivided elements. Elementsubdivider 143 can include a subdivision strategizer 144, and the degreeof freedom directory 142 can be a dynamic directory, which is updatedbetween element subdivisions. Moreover, element subdivider 143 caninclude a breadth-first-search subdivider 145 that generates nearestnewly-constrained elements and subdividing all nearest newly-constrainedelements before subdividing a neighbor of a nearest newly-constrainedelement.

The present invention reduces the number of Steiner points, and inaddition to the three-dimensional simulation mentioned above, hasfurther application in areas where reduction of Steiner points inoptimizing connection of a three-dimensional set of data points isdesirable, such as in designing pipelines for drainage systems, wiringbuildings, setting up networks within offices or larger areas such asstates or countries, connecting a number of set locations using theshortest network possible, optimizing sewer systems, power lines,transportation routes, computer assisted design for very large systemintegration (VLSI) systems, topological network design, communicationnetworks, multiprocessor scheduling, etc.

While the invention has been described in terms of its preferredembodiments, those skilled in the art will recognize that the inventioncan be practiced with modification within the spirit and scope of theappended claims.

1. A dynamic directory, stored on a computer readable medium, of degreeof freedom data for elements in a non-conformal mixed-element meshcomprising elements subdividable into tetrahedra, comprising: arespective degree of freedom value is stored for each element, whereinthe degree of freedom value is current as element subdivision proceeds.2. The directory of claim 1, wherein the element subdivision is based onthe degree of freedom values in the directory, with ordered subdivisionbeginning with relatively low degree of freedom element subdivision. 3.A tetrahedralization method, comprising at least the steps of: providinga non-conformal mixed element mesh comprising elements subdividable intotetrahedra, and identifying respective degree of freedom values for theelements in the mesh; and performing element subdivision based on thedegree of freedom values of elements in the mesh.
 4. The method of claim3, wherein element subdivision begins with a batch of relativelymost-constrained elements.
 5. The method of claim 3, wherein elementsubdivision includes look-ahead.
 6. The method of claim 3, wherein thesubdivision includes, when multiple subdivisions of an element arepossible, applying a subdivision pattern closest to satisfying Dompierre“global numbering” criteria.
 7. The method of claim 3, includingmaintaining degree of freedom data for elements in the mesh.
 8. Themethod of claim 7, including post-subdivision updating of the degree offreedom data.
 9. The method of claim 8, wherein degree of freedom datais updated after each element subdivision.
 10. The method of claim 8,wherein degree of freedom data is updated after a batch of elements havebeen subdivided.
 11. The method of claim 3, includingbreadth-first-search subdivision.
 12. The method of claim 11, whereinthe breadth-first-search subdivision includes generating nearestnewly-constrained elements and subdividing all nearest newly-constrainedelements before subdividing a neighbor of a nearest newly-constrainedelement.
 13. The method of claim 3, including obtaining tetrahedralizedoutput.
 14. Tetrahedralized output data produced by providing anon-conformal mixed element mesh comprising elements subdividable intotetrahedra, and generating data defining respective degree of freedomvalues for the elements in the mesh; and performing element subdivisionbased on the degree of freedom values of elements in the mesh, whereinthe degree of freedom data is dynamically updated.
 15. Thetetrahedralized output of claim 14, including a minimal number of, orno, Steiner points.